A branch-and-bound algorithm for the minimum radius $k$-enclosing ball problem

An Efficient Inexact Newton-CG Algorithm for the Smallest Enclosing Ball Problem of Large Dimensions

In this paper, we consider the problem of computing the smallest enclosing ball (SEB) of a set of $m$ balls in $\mathbb{R}^n,$ where the product $mn$ is large. We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special (approximate) sparsity structure. The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexa…

On a minimum enclosing ball of a collection of linear subspaces

This paper concerns the minimax center of a collection of linear subspaces. When the subspaces are $k$-dimensional subspaces of $\mathbb{R}^n$, this can be cast as finding the center of a minimum enclosing ball on a Grassmann manifold, Gr$(k,n)$. For subspaces of different dimension, the setting becomes a disjoint union of Grassmannians rather than a single manifold, and the problem is no longer well-defined. However, natural geometric maps exist between these manifolds with a well-defined notion of distance for the images of the subspaces under the mappings. Solving the initial problem in thi…

二次函数联合数值范围的一个新可分性质及其在最小包围球问题中的应用

A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem

本文研究一类二次函数联合数值范围的可分性，提出新集合$G(\Bbb R^n)^\bullet$，在秩条件为$n$时仍保持凸性，从而推动最小包围球问题的求解。

Distributed consensus on enclosing shapes and minimum time rendezvous

In this paper we introduce the notion of optimization under control and communication constraint in a robotic network. Starting from a general setup, we focus our attention on the problem of achieving rendezvous in minimum time for a network of first order agents with bounded inputs and limited range communication. We propose two dynamic control and communication laws. These laws are based on consensus algorithms for distributed computation of the minimal enclosing ball and orthotope of a set of points. We prove that these control laws converge to the optimal solution of the centralized proble…

Chebyshev Center of the Intersection of Balls: Complexity, Relaxation and Approximation

We study the n-dimensional problem of finding the smallest ball enclosing the intersection of p given balls, the so-called Chebyshev center problem (CCB). It is a minimax optimization problem and the inner maximization is a uniform quadratic optimization problem (UQ). When p<=n, (UQ) is known to enjoy a strong duality and consequently (CCB) is solved via a standard convex quadratic programming (SQP). In this paper, we first prove that (CCB) is NP-hard and the special case when n = 2 is strongly polynomially solved. With the help of a newly introduced linear programming relaxation (LP), the (SQ…

CAT(0)立方复形上的凸优化

Convex optimization on CAT(0) cubical complexes

本文研究CAT(0)立方复形上的测地凸优化问题，提出基于欧几里得切割平面算法的分解方法，用于求解最小包围球、加权均值与中位数等问题。

Variance Reduction for Matrix Games

We present a randomized primal-dual algorithm that solves the problem $\min_{x} \max_{y} y^\top A x$ to additive error $ε$ in time $\mathrm{nnz}(A) + \sqrt{\mathrm{nnz}(A)n}/ε$, for matrix $A$ with larger dimension $n$ and $\mathrm{nnz}(A)$ nonzero entries. This improves the best known exact gradient methods by a factor of $\sqrt{\mathrm{nnz}(A)/n}$ and is faster than fully stochastic gradient methods in the accurate and/or sparse regime $ε\le \sqrt{n/\mathrm{nnz}(A)}$. Our results hold for $x,y$ in the simplex (matrix games, linear programming) and for $x$ in an $\ell_2$ ball and $y$ in the s…

An ellipsoidal branch and bound algorithm for global optimization

A branch and bound algorithm is developed for global optimization. Branching in the algorithm is accomplished by subdividing the feasible set using ellipses. Lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate. A ball approximation algorithm, obtained by generalizing of a scheme of Lin and Han, is used to solve the convex relaxation of the original problem. The ball approximation algorithm is compared to SEDUMI as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal c…

Optimization Algorithms for Faster Computational Geometry

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined by $m$ hyperplanes, and the minimum enclosing ball (MinEB) of a set of $n$ points, both in $d$-dimensional space. We improve the running time of iterative algorithms on MaxIB from $\tilde{O}(m d α^3 / \varepsilon^3)$ to $\tilde{O}(md + m \sqrt{d} α/ \varepsilon)$, a speed-up up to $\tilde{O}(\sqrt{d} α^2 / \varepsilon^2)$, and MinEB from $\tilde{O}(n d / \sqrt{\varepsilon})$ to $\tilde{O}(nd + n \sqrt{d} / \sqrt{\varepsilon})$, a speed-up up to $\tilde{O}(…